A satellite dish is the shape of a paraboloid. The dish is 38 inches wide, and 10 inches deep. How many inches should the receiver be located from the vertex for optimal reception? (round to the nearest thousandth)
To find the optimal location for the receiver on a satellite dish in order to achieve the best reception, we need to determine the focal length of the parabolic shape.
The parabola is defined by the equation:
y = (1/(4 × f)) × x^2
where y represents the depth (also known as the height) of the parabola at a given horizontal distance x from the vertex, and f is the focal length.
In this case, we are given that the dish is 38 inches wide, so the horizontal distance from one side of the dish to the other is 38/2 = 19 inches.
We are also given that the depth (or height) of the dish is 10 inches.
To find the focal length f, we can substitute the given values into the equation and solve for f.
10 = (1/(4 × f)) × (19/2)^2
To simplify, we calculate (19/2)^2 = 9.5^2 = 90.25.
10 = (1/(4 × f)) × 90.25
Multiplying both sides by 4f:
40f = 90.25
Now, we isolate f by dividing both sides by 40:
f = 90.25 / 40
Simplifying:
f = 2.25625
Therefore, the focal length of the parabolic satellite dish is approximately 2.25625 inches.
To find the optimal location for the receiver, we need to place it at a distance equal to the focal length from the vertex. So, the receiver should be located approximately 2.25625 inches from the vertex for optimal reception.
the parabola x^2 = 4py has its focus at (0,p)
so set it up, using y(19) = 10